I would also go for $\Pi(t)$ or $t!$, but a possible reason to prefer the shifted version, $\Gamma(t)$, is the following. The gamma densities $\gamma_t$, $t\in \mathbb{R}$ defined as
$\gamma_t(x):=\frac{x_+^{t-1}e^{-x}}{\Gamma(t)},$
are a convolution semigroup, so that $\Gamma(t)$ appears naturally as the normalization factor of $\gamma_t$. (And, of course, the semigroup relation
$\gamma_t*\gamma_s=\gamma_{t+s}$
would be destroyed shifting from t-1 to t in the definition of $\gamma_t$)
Also note that the expression of the Beta function
$B(t,s):=\int_0^1 x^{t-1}(1-x)^{s-1}dx$
in terms of the $\Gamma$ function, if shifted, would also loose the useful form
$B(t,s)=\frac{\Gamma(t)\Gamma(s)}{\Gamma(t+s)}.$
(incidentally note that this relation follows plainly from the semigroup property since as a general fact, the integral of a convolution of two functions is the product of their integrals, it you feel like tryin the short computation)integrals).
